Answer
See the details below.
Work Step by Step
The three forms of the product rule are given as follows
(a) Scalar times vector:
$$
\frac{d}{d t}(f(t) \mathbf{r}(t))=f^{\prime}(t) \mathbf{r}(t)+f(t) \mathbf{r}^{\prime}(t).
$$
(b) Dot product:
$$
\frac{d}{d t} (r_1(t) \cdot {r_2}(t))=r_1^{\prime}(t) \cdot {r_2}(t)+r_1(t) \cdot {r_2}^{\prime}(t).
$$
(c) Cross product:
$$
\frac{d}{d t} (r_1(t) \times {r_2}(t))=r_1^{\prime}(t) \times {r_2}(t)+r_1(t) \times{r_2}^{\prime}(t).
$$
